March 2019, 24(3): 1393-1409. doi: 10.3934/dcdsb.2019021

An optimal control problem for some nonlinear elliptic equations with unbounded coefficients

Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli Federico Ⅱ, Complesso Universitario Monte Sant'Angelo, Via Cintia - 80126 Napoli, Italy

Dedicated to the memory of Prof. V. S. Melnik

Received  October 2017 Revised  March 2018 Published  January 2019

We study an optimal control problem associated to a Dirichlet boundary value problem of the type
$ \begin{equation*} \mbox{(BVP) }\,\,\,\,\,\,\,\, {\rm{div}}\; \left[ \beta(x)\nabla u (x) + \left( A\frac {x}{|x|^2 } +g(x) \right)u(x)\right] = {\rm{div}}\; \mathcal F,\quad u\in W^{1,p}_0( \Omega ) , \end{equation*} $
$ 1<p\leqslant 2, $
where
$ \Omega $
is a bounded regular domain of
$ \mathbb{R}^N $
,
$ 0\in \Omega , $
$ \beta: \Omega \rightarrow {\mathbb R} $
is an unbounded function satisfying
$ \beta(x)\geqslant\lambda_0>0 $
a.e.,
$ A $
is a suitably small constant, and
$ g\in L^\infty( \Omega ; \mathbb{R}^N ) $
.
We consider the vector field
$ \mathcal F $
as the control and the corresponding weak solution
$ u $
to (BVP) as the state. Our aim is to find the optimal vector field
$ \mathcal F\in L^p( \Omega ) $
so that the corresponding state
$ u\in W^{1,p}_0( \Omega ) $
is close to the desired profile in
$ L^p( \Omega ) $
while the norm of
$ u $
in
$ W^{1,p}( \Omega ) $
is not too large.
We prove that, for every
$ p $
less than
$ 2 $
and suitably close to
$ 2 $
, (BVP) admits an unique weak solution and for such values of
$ p $
, we prove the existence of optimal pairs.
Citation: Gabriella Zecca. An optimal control problem for some nonlinear elliptic equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1393-1409. doi: 10.3934/dcdsb.2019021
References:
[1]

A. Alvino, Sulla disuguaglianza di Sobolev in Spazi di Lorentz, Boll. Un. Mat. It. A (5), 14 (1977), 148-156.

[2] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988.
[3]

L. Boccardo, Some developments on Dirichlet problems with discontinuous coefficients, Boll. Un. Mat. It. (9), 2 (2009), 285-297.

[4]

L. Boccardo, Dirichlet problems with singular convection terms and applications, J. Differential Equations, 258 (2015), 2290-2314. doi: 10.1016/j.jde.2014.12.009.

[5]

L. Boccardo, Quelques problèmes de Dirichlet avec données dans de grands espaces de Sobolev, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 1269–1272. doi: 10.1016/S0764-4442(97)82351-8.

[6]

H. Brézis and L. Nirenberg, Degree theory and BMO-Part Ⅰ: Compact manifolds with boundary, Selecta Math., 1 (1995), 197-263. doi: 10.1007/BF01671566.

[7]

M. Carozza and C. Sbordone, The distance to $L^\infty$ in some function spaces and applications, Differential Integral Equations, 10 (1997), 599-607.

[8]

M. CarozzaG. Moscariello and A. Passarelli di Napoli, Nonlinear equations with growth coefficients in BMO, Houston Journal of Mathematics, 28 (2002), 917-929.

[9]

C. D'ApiceU. De MaioP. I. Kogut and R. Manzo, On the solvability of an optimal control problem in coefficients for ill-posed elliptic boundary value problems, Electronic Journal of Differential Equations, 2014 (2014), 1-23.

[10]

C. D'ApiceU. De Maio and P. I. Kogut, Gap phenomenon in the homogenization of parabolic optimal control problems, IMA J. Math. Control Inf., 25 (2008), 461-489. doi: 10.1093/imamci/dnn010.

[11]

U. De Maio P. Kogut and G. Zecca, On optimal $L^1$-control in coefficients for quasi-linear Dirichlet boundary value problem with $BMO$-anisotropic $p$-Laplacian, Applicable Analysis, (2019) to appear.

[12]

A. Fiorenza and C. Sbordone, Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^1$, Studia Mathematica, 127 (1998), 223-231.

[13]

F. GiannettiL. Greco and G. Moscariello, Linear elliptic equations with lower order terms, Differential and Integral Equations, 26 (2013), 623-638.

[14]

O. Giubé and A. Mercaldo, Existence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data, Potential Anal., 25 (2006), 223-258. doi: 10.1007/s11118-006-9011-7.

[15]

L. Greco and G. Moscariello, An embedding Theorem in Lorentz-Zygmund spaces, Potential Anal., 5 (1996), 581-590. doi: 10.1007/BF00275795.

[16]

L. GrecoG. Moscariello and T. Radice, Nondivergence elliptic equations with unbounded coefficients, Discrete and Continuous Dynamical Systems - Series B, 11 (2009), 131-143. doi: 10.3934/dcdsb.2009.11.131.

[17]

L. GrecoG. Moscariello and G. Zecca, Regularity for solutions to nonlinear elliptic equations, Differential and Integral Equations, 26 (2013), 1105-1113.

[18]

L. Greco, G. Moscariello and G. Zecca, An obstacle problem for noncoercive operators, Abstract and Applied Analysis, 2015 (2015), Article ID 890289, 8pp. doi: 10.1155/2015/890289.

[19]

L. Greco, G. Moscariello and G. Zecca, Very weak solutions to elliptic equations with singular convection term, Journal of Mathematical Analysis and Applications, 457 (2018), 1376–1387. doi: 10.1016/j.jmaa.2017.03.025.

[20]

T. Horsin and P. Kogut, On unbounded optimal controls in coefficients for ill-posed elliptic Dirichlet boundary value problems, Asymptotic Analysis, 98 (2016), 155-188. doi: 10.3233/ASY-161365.

[21]

T. Horsin and P. Kogut, Optimal $L^2$-control problem in coefficients for a linear elliptic equation. Ⅰ. Existence result, Mathematical Control and Related Fields, 5 (2015), 73–96. doi: 10.3934/mcrf.2015.5.73.

[22]

T. HorsinP. Kogut and O. Wilk, Optimal $L^2$-control problem in coefficients for a linear elliptic equation. Ⅱ. Approximation of solutions and optimality conditions, Mathematical Control and Related Fields, 6 (2016), 595-628. doi: 10.3934/mcrf.2016017.

[23]

T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math., 454 (1994), 143-161. doi: 10.1515/crll.1994.454.143.

[24]

N. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-426. doi: 10.1002/cpa.3160140317.

[25]

H. Kim and Y. Kim, On weak solutions of elliptic equations with singular drifts, SIAM J. Math. Anal., 47 (2015), 1271-1290. doi: 10.1137/14096270X.

[26]

P. Kogut, On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 34 (2014), 2105-2133. doi: 10.3934/dcds.2014.34.2105.

[27]

O. P. Kupenko, Optimal control problems in coefficients for degenerate variational inequalities of monotone type. Ⅰ. Existence of optimal solutions, J. Computational and Appl. Mathematics, 106 (2011), 88-104.

[28]

O. P. Kupenko, Optimal control problems in coefficients for degenerate variational inequalities of monotone type. Ⅱ. Attainability problem, J. Computational and Appl. Mathematics, 107 (2012), 15-34.

[29]

A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces, Comm. Math. Univ. Carolinae, 25 (1984), 537-554.

[30]

O. P. Kupenko and R. Manzo, On optimal controls in coefficients for ill-posed nonlinear elliptic Dirichlet boundary value problems, Discrete and Continuous Dynamical Systems, Series B, 23 (2018), 1363-1393. doi: 10.3934/dcdsb.2018155.

[31]

J. L. Lions, Optimal Control of System Governed by Partial Differential Equations, Springer, Berlin, 1971.

[32]

S. Monsurrò and M. Transirico, Noncoercive elliptic equations with discontinuous coefficients in unbounded domains, Nonlinear Analysis, 163 (2017), 86-103. doi: 10.1016/j.na.2017.07.008.

[33]

G. Moscariello, Existence and uniquesness for elliptic equations with lower-order terms, Adv. Calc. Var., 4 (2011), 421-444. doi: 10.1515/ACV.2011.007.

[34]

R. O'Neil, Integral transforms and tensor products on Orlicz spaces and $L(p, \, q)$ spaces, J. Analyse Math., 21 (1968), 1-276.

[35]

T. Radice and G. Zecca, Existence and uniqueness for nonlinear elliptic equations with unbounded coefficients, Ricerche di Matematica, 63 (2014), 355–367. doi: 10.1007/s11587-014-0202-z.

[36]

J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa (3), 18 (1964), 385-387.

[37]

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier, 15 (1965), 189-258. doi: 10.5802/aif.204.

[38]

B. Stroffolini, Elliptic systems of PDE with BMO coefficients, Potential analysis, 15 (2001), 285-299. doi: 10.1023/A:1011290420956.

[39]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.

[40]

G. Zecca, Existence and uniqueness for nonlinear elliptic equations with lower-order terms, Nonlinear Analysis, 75 (2012), 899-912. doi: 10.1016/j.na.2011.09.022.

[41]

M. Zgurovsky, V. S. Mel'nik and P. O. Kasyanov, Evolution Inclusions and Variation Inequalities for Earth Data Processing I. Operator Inclusions and Variation Inequalities for Earth Data Processing, Advances in Mechanics and Mathematics, 24. Springer-Verlag, Berlin, 2011.

[42]

V. V. Zhikov and S. E. Pastukhova, Operator estimates in homogenization theory, Russian Math. Surveys, 71 (2016), 417-511. doi: 10.4213/rm9710.

[43]

V. V. Zhikov, Remarks on the uniqueness of the solution of the Dirichlet problem for a second-order elliptic equation with lower order terms, Funct. Anal. Appl., 38 (2004), 173-183. doi: 10.1023/B:FAIA.0000042802.86050.5e.

show all references

References:
[1]

A. Alvino, Sulla disuguaglianza di Sobolev in Spazi di Lorentz, Boll. Un. Mat. It. A (5), 14 (1977), 148-156.

[2] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988.
[3]

L. Boccardo, Some developments on Dirichlet problems with discontinuous coefficients, Boll. Un. Mat. It. (9), 2 (2009), 285-297.

[4]

L. Boccardo, Dirichlet problems with singular convection terms and applications, J. Differential Equations, 258 (2015), 2290-2314. doi: 10.1016/j.jde.2014.12.009.

[5]

L. Boccardo, Quelques problèmes de Dirichlet avec données dans de grands espaces de Sobolev, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 1269–1272. doi: 10.1016/S0764-4442(97)82351-8.

[6]

H. Brézis and L. Nirenberg, Degree theory and BMO-Part Ⅰ: Compact manifolds with boundary, Selecta Math., 1 (1995), 197-263. doi: 10.1007/BF01671566.

[7]

M. Carozza and C. Sbordone, The distance to $L^\infty$ in some function spaces and applications, Differential Integral Equations, 10 (1997), 599-607.

[8]

M. CarozzaG. Moscariello and A. Passarelli di Napoli, Nonlinear equations with growth coefficients in BMO, Houston Journal of Mathematics, 28 (2002), 917-929.

[9]

C. D'ApiceU. De MaioP. I. Kogut and R. Manzo, On the solvability of an optimal control problem in coefficients for ill-posed elliptic boundary value problems, Electronic Journal of Differential Equations, 2014 (2014), 1-23.

[10]

C. D'ApiceU. De Maio and P. I. Kogut, Gap phenomenon in the homogenization of parabolic optimal control problems, IMA J. Math. Control Inf., 25 (2008), 461-489. doi: 10.1093/imamci/dnn010.

[11]

U. De Maio P. Kogut and G. Zecca, On optimal $L^1$-control in coefficients for quasi-linear Dirichlet boundary value problem with $BMO$-anisotropic $p$-Laplacian, Applicable Analysis, (2019) to appear.

[12]

A. Fiorenza and C. Sbordone, Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^1$, Studia Mathematica, 127 (1998), 223-231.

[13]

F. GiannettiL. Greco and G. Moscariello, Linear elliptic equations with lower order terms, Differential and Integral Equations, 26 (2013), 623-638.

[14]

O. Giubé and A. Mercaldo, Existence and stability results for renormalized solutions to noncoercive nonlinear elliptic equations with measure data, Potential Anal., 25 (2006), 223-258. doi: 10.1007/s11118-006-9011-7.

[15]

L. Greco and G. Moscariello, An embedding Theorem in Lorentz-Zygmund spaces, Potential Anal., 5 (1996), 581-590. doi: 10.1007/BF00275795.

[16]

L. GrecoG. Moscariello and T. Radice, Nondivergence elliptic equations with unbounded coefficients, Discrete and Continuous Dynamical Systems - Series B, 11 (2009), 131-143. doi: 10.3934/dcdsb.2009.11.131.

[17]

L. GrecoG. Moscariello and G. Zecca, Regularity for solutions to nonlinear elliptic equations, Differential and Integral Equations, 26 (2013), 1105-1113.

[18]

L. Greco, G. Moscariello and G. Zecca, An obstacle problem for noncoercive operators, Abstract and Applied Analysis, 2015 (2015), Article ID 890289, 8pp. doi: 10.1155/2015/890289.

[19]

L. Greco, G. Moscariello and G. Zecca, Very weak solutions to elliptic equations with singular convection term, Journal of Mathematical Analysis and Applications, 457 (2018), 1376–1387. doi: 10.1016/j.jmaa.2017.03.025.

[20]

T. Horsin and P. Kogut, On unbounded optimal controls in coefficients for ill-posed elliptic Dirichlet boundary value problems, Asymptotic Analysis, 98 (2016), 155-188. doi: 10.3233/ASY-161365.

[21]

T. Horsin and P. Kogut, Optimal $L^2$-control problem in coefficients for a linear elliptic equation. Ⅰ. Existence result, Mathematical Control and Related Fields, 5 (2015), 73–96. doi: 10.3934/mcrf.2015.5.73.

[22]

T. HorsinP. Kogut and O. Wilk, Optimal $L^2$-control problem in coefficients for a linear elliptic equation. Ⅱ. Approximation of solutions and optimality conditions, Mathematical Control and Related Fields, 6 (2016), 595-628. doi: 10.3934/mcrf.2016017.

[23]

T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math., 454 (1994), 143-161. doi: 10.1515/crll.1994.454.143.

[24]

N. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-426. doi: 10.1002/cpa.3160140317.

[25]

H. Kim and Y. Kim, On weak solutions of elliptic equations with singular drifts, SIAM J. Math. Anal., 47 (2015), 1271-1290. doi: 10.1137/14096270X.

[26]

P. Kogut, On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients, Discrete and Continuous Dynamical Systems - Series A (DCDS-A), 34 (2014), 2105-2133. doi: 10.3934/dcds.2014.34.2105.

[27]

O. P. Kupenko, Optimal control problems in coefficients for degenerate variational inequalities of monotone type. Ⅰ. Existence of optimal solutions, J. Computational and Appl. Mathematics, 106 (2011), 88-104.

[28]

O. P. Kupenko, Optimal control problems in coefficients for degenerate variational inequalities of monotone type. Ⅱ. Attainability problem, J. Computational and Appl. Mathematics, 107 (2012), 15-34.

[29]

A. Kufner and B. Opic, How to define reasonably weighted Sobolev spaces, Comm. Math. Univ. Carolinae, 25 (1984), 537-554.

[30]

O. P. Kupenko and R. Manzo, On optimal controls in coefficients for ill-posed nonlinear elliptic Dirichlet boundary value problems, Discrete and Continuous Dynamical Systems, Series B, 23 (2018), 1363-1393. doi: 10.3934/dcdsb.2018155.

[31]

J. L. Lions, Optimal Control of System Governed by Partial Differential Equations, Springer, Berlin, 1971.

[32]

S. Monsurrò and M. Transirico, Noncoercive elliptic equations with discontinuous coefficients in unbounded domains, Nonlinear Analysis, 163 (2017), 86-103. doi: 10.1016/j.na.2017.07.008.

[33]

G. Moscariello, Existence and uniquesness for elliptic equations with lower-order terms, Adv. Calc. Var., 4 (2011), 421-444. doi: 10.1515/ACV.2011.007.

[34]

R. O'Neil, Integral transforms and tensor products on Orlicz spaces and $L(p, \, q)$ spaces, J. Analyse Math., 21 (1968), 1-276.

[35]

T. Radice and G. Zecca, Existence and uniqueness for nonlinear elliptic equations with unbounded coefficients, Ricerche di Matematica, 63 (2014), 355–367. doi: 10.1007/s11587-014-0202-z.

[36]

J. Serrin, Pathological solutions of elliptic differential equations, Ann. Scuola Norm. Sup. Pisa (3), 18 (1964), 385-387.

[37]

G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier, 15 (1965), 189-258. doi: 10.5802/aif.204.

[38]

B. Stroffolini, Elliptic systems of PDE with BMO coefficients, Potential analysis, 15 (2001), 285-299. doi: 10.1023/A:1011290420956.

[39]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.

[40]

G. Zecca, Existence and uniqueness for nonlinear elliptic equations with lower-order terms, Nonlinear Analysis, 75 (2012), 899-912. doi: 10.1016/j.na.2011.09.022.

[41]

M. Zgurovsky, V. S. Mel'nik and P. O. Kasyanov, Evolution Inclusions and Variation Inequalities for Earth Data Processing I. Operator Inclusions and Variation Inequalities for Earth Data Processing, Advances in Mechanics and Mathematics, 24. Springer-Verlag, Berlin, 2011.

[42]

V. V. Zhikov and S. E. Pastukhova, Operator estimates in homogenization theory, Russian Math. Surveys, 71 (2016), 417-511. doi: 10.4213/rm9710.

[43]

V. V. Zhikov, Remarks on the uniqueness of the solution of the Dirichlet problem for a second-order elliptic equation with lower order terms, Funct. Anal. Appl., 38 (2004), 173-183. doi: 10.1023/B:FAIA.0000042802.86050.5e.

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